Modular-invariant large-N completion of an integrated correlator in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 supersymmetric Yang-Mills theory

The use of supersymmetric localisation has recently led to modular covariant expressions for certain integrated correlators of half-BPS operators in N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 4 supersymmetric Yang-Mills theory with a general classical gauge group GN. Here we determine generating functions that encode such integrated correlators for any classical gauge group and provide a proof of previously conjectured formulae. This gives a systematic understanding of the relation between properties of these correlators at finite N and their expansions at large N. In particular, it determines a duality-invariant non-perturbative completion of the large-N expansion in terms of a sum of novel non-holomorphic modular functions. These functions are exponentially suppressed at large N and have the form of a sum of contributions from coincident (p, q)-string world-sheet instantons.